Question: Simplify; express your answer in exponential form. Assume $t\neq 0, z\neq 0$. $\dfrac{{(t^{2}z^{5})^{-5}}}{{(t^{-4}z^{-5})^{-3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{2}z^{5})^{-5} = (t^{2})^{-5}(z^{5})^{-5}}$ On the left, we have ${t^{2}}$ to the exponent ${-5}$ . Now ${2 \times -5 = -10}$ , so ${(t^{2})^{-5} = t^{-10}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{2}z^{5})^{-5}}}{{(t^{-4}z^{-5})^{-3}}} = \dfrac{{t^{-10}z^{-25}}}{{t^{12}z^{15}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-10}z^{-25}}}{{t^{12}z^{15}}} = \dfrac{{t^{-10}}}{{t^{12}}} \cdot \dfrac{{z^{-25}}}{{z^{15}}} = t^{{-10} - {12}} \cdot z^{{-25} - {15}} = t^{-22}z^{-40}$